Guido "Wugi" Wuyts @ Dilbeek, Belgium, Europe, World, Solar System, Milky Way, Local Cluster, ...

As a youth I developed this theory which I called of "hyperreal" numbers, to encompass complex numbers, quaternions and alike. Let me hazard a simple summary here:

n-dim Numbers as n x n matrices

Define a Vector space upon an n-variant family of square n x n matrices as follows :

Let any matrix Q(ij) of the family be generated as a vector q(m) upon n independent units E(m,ij) :

                Q = q_m . E_m           , or in matrix notation:

                Q_ij = q_m . E_mij

As a first requirement, the family should include the product of two members of the family.

The product writes:

                Q * R = T             , or

                Q_ij . R_jk = T_ik = q_m . r_n . E_mij . E_njk,

and would be a member if

                T_ik = t_m . E_mik

Therefore, multiplying the units ought to yield a member itself:

                E_m * E_n = F_mns . E_s    (1) , or

                E_mij . E_njk = F_mns . E_sik                 (1')  (for all  m i n k)

Now, if we wish the first column of the matrices to represent the vector variables qi on either row i (the first column represents the “coordinate”), we should have:

                Q_i1 = q_m . E_mi1 = q_i   , or

                E_mi1 = δ_mi                                 (which = 0 for m ≠ i, and = 1 for m = i)

Applying this to the previous equation (1') , for k=1, we obtain

                E_mij . E_nj1 = F_mns . E_si1,

                E_mij . δ_nj = F_mns . δ_si                        , so

                E_min = F_mni

Redefining (1) with F-factors, we get

                F_mji . F_nkj = F_mns . F_ski                 (1'bis)  

I called these the "Autovariance conditions", assuring that the matrix family embraces any product of its members. Without these conditions, a random product would “leave” the n-dim matrix family into the n x n matrix space!

Now, for the family to be a Field, it would have to offer a single "inverse member" for all non-zero members, i.e. there should exist for each Q, an R that produces

                Q * R = E1                                          , or

                Q_ij . R_jk = E_1ik

for k=1:  Q_ij . R_j1 = E_1i1                                       , or

                (q_m . E_mij) . r_j = δ_i1                (2)

This system of equations should be solvable to rj, for all non-zero qm.

Therefore, the determinant of the system matrix should be a positive-definite polynomial of the qm:

As a general result, only even values of dimension n are possible, as only even-powered polynomials might be positive-definite.

This second condition taken alone would result in what I called an "Abstract Field".

So, Abstract Fields would not be required to comply with the Autovariance condition. The matrix product would then not be itself a member of the matrix n-dim family, or, the operation rules (1) on the units apply to the units, but not to their matrices (1'). The n-vector matrices are not to be seen as fully representative for the Abstract Field, as their matrix product Em ° En is not internal in the n-family, only the (abstract) operation Em * En is.

Abstract Fields that DO comply to the autovariance condition I called "Hyperreal Fields", as they seem a proper extension of the real numbers (rather than the complex ones, as suggested by the term "hypercomplex") with respect to an internal product.

The autovariance conditions   

                F_mji . F_nkj = F_mns . F_ski                 (1'bis)

They form a system of equations with following characteristics:

                -   4 indices m, i, n, k , so n⁴ equations

                -   quadratic in the F

                -   not independent

                -   an unknown number of freedom degrees (or dependent equations)

                     (the definiteness of the determinant may add restrictions to the freedom degrees:

                     positive, range..., see examples)

It is impossible to solve such system analytically, but let us get an idea of the numbers involved. First, the non-trivial equations. Since we're dealing with matrices, we can require that E1 represent the true identity, so

                E_1kj = F_1jk = δ_jk         (besides E_mi1 = F_m1i = d_mi)                   (3)

For m=1 , n=1 , k=1, the equations become trivial. The number of non-trivial equations is:

                NT = n⁴                                  (4 arbitrary indices)

                                - (3 n³                     (one of 3 indices = 1)

                                - (3 n²                     (two of 3 indices = 1)

                                - n))                        (all 3 indices = 1)

                NT = n⁴ – 3 n³ + 3n² –n                  =  n (n³ – 3n² + 3n – 1)

                NT = n (n -1)³

As for the number of variables F, given the conditions (3), and calculating in a similar way,

                NF = n³ – 2 n² + n                 variables, or

                NF = n (n -1)²

The max number of independent equations is NF.

The min number of dependent equations is

                NT – NF = n (n – 1)² (n – 2)

Theorem

The autovariance conditions are equivalent with an associative * operation of an Abstract Field

Let           E_m * (E_n * E_k)        = (E_m * E_n) * E_k

                E_m * (F_nkj . E_j)          = F_mns . E_s * E_k

                F_mji . F_nkj . E_i            = F_mns . F_ski . E_i     , or, per base unit E_i :

                F_mji . F_nkj                 = F_mns . F_ski                        

These are the autovariance conditions (1'bis)

An Abstract Field with an associative * operation (and an identity unit E₁) can be identified with a Hyperreal Field.

Some Field families

 1.   2-dim Abstract Fields

       1A.  Operation rules (b, r  real non zero; a  real; t  real within the range -1 to +1) :

                 E₁ * E₁ = E₁

                 E₂ * E₁ = E₂

                 E₁ * E₂ = a E₁ + b E₂

                 E₂ * E₂ = -r² b E₁+ (a + 2tbr) E₂

       1B.  Alternative operation rules (see also further on, about conjugate fields) :

                 E₁ * E₁ = E₁

                 E₂ * E₁ = E₂

                 E₁ * E₂ = a E₁ + - r² c E₂

                 E₂ * E₂ = c E₁+ (a + 2tcr) E₂

Theorem

 1A + 1B generate all 2 x 2 matrices 

 If the Abstract Field under 1A is written as

                 AF = AF(a, b, r², tr)

 then the field under 1B may be written as

                 AF' = AF(a, -r² b, (1/r)², -t/r)

 The corresponding (but not autovariant) matrices can be developed from

                 Q_ij = q_m . E_mij = q_m . F_mji

 We get, in AF:

                 q = x . E₁ + y . E₂=                ┌                x                ax + by                                ┐

                                                                └                y                -r² bx + (a+2tbr)y                ┘

 and, in AF' :

                 t' = u . E₁' + v . E₂' =               ┌                u                au – r² bv                           ┐

                                                                └                v                bu + (a+2tbr)v                    ┘

 It suffices to prove that any matrix

                 ┌                A                C                ┐

                 └                B                D                ┘

 can be written as   q + t'   for some   x,y,u,v

 (4 Cramer equations in variables x,y,u,v with "equal to" parameters A,B,C,D.

 Prove that the determinant Det = f(a,b,r,t) = b²(r²+1)² ≠ 0)

 Remark.

 This theorem holds only for dim n=2, as only then can an  n+n-system  generate an  n²-space.

2.   2-dim Hyperreal Fields

       See 1A. , with a=0, b=1, t, r,

       yielding Q = x E₁ + y E₂ with matrices

      ┌                Q₁₁                        Q₁₂               ┐      =         ┌                x                -r2 y          ┐

      └                Q₂₁                        Q₂₂               ┘                 └                y          x + 2tr y           ┘

      Complex numbers have r = 1, t = 0.

3.   A 4-dim Hyperreal Field   

      might be constructed by considering the complex number matrix

                ┌                A                -B'               ┐

                └                B                A'                ┘

      which is autovariant and has a positive definite determinant

                P(A, B) = AA’ + BB’ ,

      and writing A = x+iy, B= u+iv, A'=x-iy, B'=u-iv themselves as matrices according to 2.

      We get

                ┌                x               -y                -u                -v              ┐

                │                y                x                v                -u                │

                │                u                -v                x                y                │

                └                v                 u                -y                x               ┘

      This should be something (un)like quaternions.

n-dim Numbers as geometrical transformations in n-dim space

My original approach was actually a mix of discovering Abstract and Hyperreal Field features. It started as an extension to n-dim space of my little physical theory on the real numbers themselves, where I viewed the reals as, what I called, free transformations of the straight vector line. It all boils down to defining a real number r as the universal transformation, upon a random straight line equiped with a random unit vector e, yielding:

r: e --> f = r . e 

It may seem odd to want to define the reals upon a "free" vector space, whereas these structures are usually defined themselves using the reals, but it kinda worked, and seemed a good way to make a statement on the deep similitude between the reals and the straight line (I have since become more sceptic about this similitude:‑)

I thought then of generalising the notion of reals to n-dim numbers Q=q_i . E_i , to be defined as free transformations upon an n-dim Euclidean space with base e_j, yielding:

Q: e₁ --> q = q_i . e_i

E_j: e₁ --> δ_ij . e_i = e_j

and equiped with an internal operation

E_i * E_j = F_ijk . E_k

But hence the picture starts getting complicated, leaving various degrees of freedom, eg:

How to relate the n-dim transform family to the n x n-dim matrix transform family?

In particular, how to define the further base transforms:

E_j: e_k --> ? , or

E_j ° E_k: e₁ --> ?

E_j ° E_k: e_m --> ?

Relationship between * (internal in base E_i) and ° (generally, not internal in base E_i) ?

Is E₁ unit for * , or not? (it is for °)  In other words: is generally

E_i * E₁     = F_i1k . E_k                  (which I called "General Abstract Fields")   , or specifically

E_i * E₁     = E_i                         ("Ordinary Abstract Fields") ?

I had to allow for General Abstract Fields in the light of the following theorem.

Theorem on Conjugate Abstract Fields

If the parameters Fijk  define an Abstract Field for the operation * :

E_i * E_j = F_ijk . E_k

then also do the conjugate parameters F_jik , F_ikj , F_jki , F_kij , F_kji .                (3) 

Let q = q_i . E_i ≠ 0. Then a solution r = r_j . E_j ≠ 0 should exist to the equation

                q * r = E₁ , or

                q_i . r_j . E_i * E_j = E₁

                q_i . r_j . F_ijk . E_k = δ_1k . E_k , or

                (q_i . F_ijk) . r_j = δ_1k     (compare 2)

This can be written as

                (r_j . F_ijk) . q_i = δ_1k  

Now, not to allow a non-zero q occurring without a non-zero solution r, means that the determinant

                P'(r(m)) = det(r_j . F_ijk)                 cannot be zero. This can be rewritten

                P'(r(m)) = det(r_i . F_jik)                 , so

                F'_ijk = F_jik                 (3')

define another set of valid operation rules.

Furthermore, considering that the determinant of a transposed matrix remains unchanged, the other permutations are also proved.

This leaves us with what I called six Conjugate Abstract Fields. It may be that some of them are congruent of course.

4.   General 2-dim Abstract Fields

Given the general operation rules

                E₁ * E₁ = α . E₁ + β . E₂                         E₁ * E₂ = a . E₁ + b . E₂

                E₂ * E₁ = γ . E₁ + δ . E₂                          E₂ * E₂ = c . E₁ + d . E₂ 

which may be written down in a (*) table: 

or, shorthand:

Setting out to find an inverse (u,v) for a number (x,y), thus

                (x . E₁ + y . E₂) * (u . E₁ + v . E₂) = E₁

Stating that the determinant P(x,y) of the equations in (u,v) must be positive definite

                P(x,y) = (αb-aβ) x² + (αd-aδ + γb-cβ) xy + (γd-cδ) y²

Conjugate conditions hold for the determinant P'(u,v) of the equations in (x,y)

A result is that all six "surface determinants" of the tensor formed by

                α                β                and                 a                b

                γ                δ                                       c                d

must differ from zero.

The coefficients of P(x,y) yield conditions:

                (α b-a β) = + r²             or = – r²

                (γ d-c δ) = + s²              or = – s²

                (α d-a δ + γ b-c β) = 2 trs                (r, s non-zero reals, abs(t) < 1)

which can be resolved, eg, to  β, γ and δ  for non-zero  a and c  and for any  b and  d.

As an example, two families (+ and -) of six conjugate Abstract Fields are obtained from an initial choice

                a=1, b=2, c=3, d=4, α=5, r=2, s=3, t=0 :

We obtain

                β =                 6                   OR                14

                γ =                 -7.5               OR                37.5

                δ =                 -13                OR                 53

Operation rules:

5.   Some 4-dim Hyperreal and Abstract Fields

Let us assume a number

                Q = X . E1 + Y . E2 + Z . E3 + U . E4   

or, in matrix form:

Autovariance conditions give, for

                E2 * E2 :                 a1 = a6 . a9

                E3 * E3 :                 a4 = a3 . a8

                E4 * E4 :                 a7 = a2 . a5

We can rewrite the matrix :

More autovariance conditions:

                ab = ec                   =>           a²bd = c²ef         =>            a² = c²

                ad = cf                                    b²ad = e²cf                         b² = e²

                ae = bc                                   d²ab = f²ce                         d² = f²

                af = cd

                bd = ef

                bf = ed

The signs of a/c, b/e and d/f must be the same for autovariance. This leaves two solutions. Moreover, the determinant of the matrix must be positive definite. This leaves one valid solution, respecting the conditions

                ab < 0 ; ad > 0 ; bd < 0

yielding again two Hyperreal Field families :

                Q1 :                 a = r²,                 b = - s²,                 d = t²                      , and     

                Q2 :                 a = - r²,                 b = s²,                 d = - t²

so:

and:

The operation rules for both families become:   

Finally, swapping the E_i for X, Y, Z, U and vice versa, applying the conjugates theorem, (1) and (2) become matrices that define two Abstract Field families (not Hyperreals) with operation rules Q1 and Q2.

Further scope for examination

Considering an n-dim number

                Q = q_i . E_i

as a transformation of a base vector e₁ to a point

                Q(e₁) = q_i . e_i                 , but also of any base vector e_j to

                Q(e_j) = Q_ij . e_i      ,

one could wonder how Q would act on new base vectors, after a base transform. Generally, autovariance would not be conserved, unless the transform itself corresponds precisely with an n-dim number R, and we're left with an "ordinary" product operation:

Be           e'_k = R(e_k) = R_jk . e_j

then        Q(e'_k) = Q(R(e_k)) = Q*R(e_k) = Q_ij R_jk . e_i

preserves autovariance. 

Another field for exploration is 6-dim numbers. They cannot be made up from complex 3 x 3 matrices, because these would not make sure for a positive definite determinant, as was the case for dim 4.  So, the 6-dim numbers would not readily be an extension of complex numbers, but "really" of the reals. At least, if they are found (to wit: a 6‑dim matrix form with a positive definite determinant).

A matter of concern is the redundant autovariance conditions. I generously referred to them as dependent equations, but could it be that with increasing n one runs into systems with essentially incompatible equations? Could there be a restriction on the extensibility of dimensions lurking here?

Other ideas, some freer, some more restrictive, are met with in other sources, see next.

Other sources on the Net  

When looking around on the web I found out that a good deal of "my" Hyperreals is being described by what are commonly called hypercomplex numbers. I regret to see that "hyperreal" is used for a family of the reals plus the infinitesimals plus infinite numbers. So, I'd have to leave the term "hyperreal" for "hypercomplex".

Moreover, these are commonly thought of as a direct extension of the complex numbers, rather than of the reals as I did. Moreover yet, it is argued that only a few such families should exist, of dimension 2^N, each next N loosing some field characteristics.

One limiting feature is the norm. It is true that my Hyperreals have no obvious norm. You can build one of course from the q(m) , but the norm of a product is not a product of the norms. Another "drawback" is being non-commutative. Another yet, being non-associative (as is said of 16-dim sedonions; I wonder about my Hyperreals for increasing n...).

Anyway, putting restrictions may be interesting, dismissing them may also be.

First, my E1 is not entirely an identity unit, as for Abstract Fields I allow E1* Ej ≠ Ej ≠ Ej * E1 (it is only a divisor unit). So that I'm even not talking Groups ;-)

Elsewhere, I did come across base operation descriptions of type (1) (seemingly ignoring norms), but not really investigating the F-factors (their tensor aspect, their relating vector to matrix components, their autovariance) nor telling how, given such general relations, one ends up reducing the number of recognised algebras so drastically.

In yet other places I discovered examples of Rings which allow for zero divisors. In the case of my Hyperreal Fields we might want abandoning the dividability condition, and get to examine odd-dimensional rings (3-dim for a start) on autovariance...

Here are some interesting sites:

Algebras over a field
http://en.wikipedia.org/wiki/Algebra_over_a_field
Contains almost textually the relations (1)

In disguise (with the usual unit i 's) they also appear in :
http://math.hyperjeff.net/hypercomplex/hyper.html
mentioning 36 coefficients to be established for quaternions.

The common use of the term "hyperreal number"
http://en.wikipedia.org/wiki/Hyperreal_number
So, who gets me an alternative for mine ? :-)

Hypercomplex numbers
http://en.wikipedia.org/wiki/Hypercomplex_number
i x i = 0  allowed
why limit the possibilities for i x i  to  -1, 0, 1 ?
why only dim 2^N ?

http://www.hypercomplex.us/docs/generalized_number_system.pdf
Example of a 3-dim ring (Tom Jewitt's page mentioned in previous link: not found?...)

Quaternions
http://mathworld.wolfram.com/Quaternion.html
Matrix representation with a negative coefficient in first column

http://en.wikipedia.org/wiki/Quaternion
Matrix (with positive coefficients in first column) different from previous, and from mine

Commutative Hypercomplex Mathematics
http://home.usit.net/~cmdaven/hyprcplx.htm
Unlike non-commutative quaternions, a 4-dim ring with two zero divisor planes (not found ?...)

Conclusion

In many aspects my constructs of Abstract and Hyperreal Fields have been treated before and elsewhere, and would not seem original (though I developed them by myself, as a youth peccadillo:-)

Still, I'm missing in the other sources

- the thorough treatment of the internal operation (1)

- the tensor character of the F-factors and, in particular, the occurrence of (six) conjugates

- the relation between a number's vector q and matrix Q through these F-factors

- systematicity in the matrix representation (negative coefficients in the first column, instead of exposing the q-vector)

- the relationship between the n-dim number Field, the n x n operation Space it belongs to, and the n-dim vector Space it acts upon

- the autovariance property of n-variant matrix families (in  n x n space)